It is known that any natural number N may be represented as follows: ##EQU1## where n is the digit capacity of the code, and .phi..sub.p (B) is a certain Fibonacci p-number.
Fibonacci p-numbers are determined with p.gtoreq.0, which is an integer, by the following recurrence relation: ##EQU2##
With p=0, Fibonacci p-codes are a generalization of the classical binary method of number representation; with p=.infin., they coincide with what is known as the "unitary" code (cf. A. P. Stakhov, "Vvedeniye v algoritmicheskuyu teoriyu izmereniy"/"Introduction to the Algorithmic Theory of Measurements"/, Sovietskoye Radio Publishers, Moscow, 1977).
The "golden" p-code signifies a higher level of notation. The "golden" p-code of a real number A is the following method whereby A is represented as a sum total of degrees of the "golden" p-proportion where p.epsilon.{1, 2, 3, . . . , .infin.}: ##EQU3## where .sub.B.sup..epsilon. {0, 1B} is a binary number in the Bth digit of the "golden" p-code; .sub.p.sup.B is the weight of the Bth digit (i.e. the Bth degree of the "golden" p-proportion); .sub.p.sup.B is the "golden" proportion which is the real root of this equation: EQU X.sup.p+1 =X.sup.p +1. (4)
With p=1, ##EQU4## The coefficient ##EQU5## is referred to as the "golden" proportion; hence, the name of the code (3).
The "golden" proportion .sub.p possesses this fundamental property: EQU .sub.p.sup.n = .sub.p.sup.n-1 + .sub.p.sup.n-p-I, (5)
which follows directly from (4).
Fibonacci p-codes and "golden" p-codes are irrational-base codes, since ##EQU6##
The basic distinguishing feature of irrational-base codes is their redundancy, which means that each number A has several representations in irrational-base codes. For example, with p=1 , the number 8 can be represented by the following Fibonacci I-code (p=1):
______________________________________ Weights of Digits 13 8 5 3 2 1 1 ______________________________________ Fibonacci 1-code 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 1 0 = 8 0 0 1 0 1 0 1 ______________________________________
In the "golden" I-code:
__________________________________________________________________________ Weights of Digits ##STR1## ##STR2## ##STR3## ##STR4## ##STR5## ##STR6## ##STR7## ##STR8## ##STR9## __________________________________________________________________________ 1 0 0 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 1 = 8 0 1 0 1 1 1 1 0 1 __________________________________________________________________________
An important concept of the theory of irrational-base codes is the concept of the normal, or minimal, form. This is to be understood as an irrational-base code of a number A, wherein any group of p+1 successive code digits does not have more than one units digit. The process of reducing an irrational-base code to a minimal form is referred to as normalization of an irrational-base code. With p=1, the normalization is carried out by performing all operations involved in the convolution of binary digits of the irrational-base code. The convolution of binary units digits a.sub.n-1 and a.sub.n-2 to the zero digit a.sub.n =0 is to be understood as a replacement of the digit values by the respective negative values, i.e. ##EQU7##
The operation of convolution is designated as ##EQU8## The reverse operation is referred to as devolution of a digit and designated as ##EQU9## It must be emphasized that the operations of convolution and devolution of binary digits do not alter the number A represented by a code because of the fundamental feature expressed in (5).
The convolution of the "golden" code comprises performing all the operations involved in the convolution of binary digits. Here is an example for the golden proportion code: ##EQU10##
The devolution of the "golden" code consists in performing all the operations involved in the devolution of the binary digits.
For example, for the "golden" proportion code:
______________________________________ ##STR10##
______________________________________